D C A B E Neveah’s Claim • The total angle measure T of the interior and exterior angles of any polygon is T = n • 180, where n is the number of angles. • So the sum S of the interior angles of any polygon is that total minus the exterior angles. S = (n • 180°) − 360° A B C 180° 180° 180° Nic’s Claim I think exterior angles are like “walking around” a polygon. I use this idea to show that the sum of the interior angles of a triangle is 180°. • The sum of the supplementary angles is 3 • 180° = 540°. • I subtract the sum of the exterior angles, which is 360°. • This leaves 180° for the sum of the interior angles of the triangle. Situation B. Quadrilaterals and Algebra The following quadrilateral is formed by the intersection of four lines. The measures of some angles are given. Note that none of the lines are parallel to each other. 60° 70° 75° x w y z 1. Find the measures of the angles represented by w, x, y, and z. Write equations to help show your reasoning. Students may reason in different orders. One solution might be: ∠x is vertical to the 70° angle. So ∠x = 70°. ∠w and ∠60° are supplementary. ∠w + ∠ 60° = 180° That makes ∠w = 120°. 3.3 Answers Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles 189 SAMPLE
We know the measure of angles x and w. We know ∠x + ∠w + ∠y + 75° = 360. So 70° + 120° + ∠y + 75° = 360 ∠y = 360 − 70 − 120 − 75 ∠y = 95° 2. Are any of the labeled angles vertical? Supplementary? Complementary? Explain why. Pairs of Vertical Angles Pairs of Supplementary Angles Pairs of Complementary Angles angle 70° and x y and z Note that there are 16 possible pairs of vertical angles but that only two are labeled on the drawing. y and z w and angle 60° Each pair of angles adds up to 180°. Note that there are many possible pairs of supplementary angles but that only two are labeled on the drawing. None of the given (labeled) angles are complementary (add up to 90°). NOW WHAT DO YOU KNOW? What do you know about the exterior angles of a polygon? How might this knowledge be useful? Exterior angles are the supplement of interior angles. Knowing that the sum of the two angles is 180° can help us solve problems about the angles on a polygon. The sum of the interior angles of polygons changes depending on the number of sides. The sum of the exterior angles of any polygon is always 360° 3.3 Answers 190 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles 191 Name Date Class D C A B E Pentagonal Path 3.3 LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
At a Glance The objective of this problem is to develop student understanding of the conditions that determine unique shapes. This is the foundation for future work on similarity and congruence. The storyline is a challenge to convey a lot of information with only a few words to produce a unique triangle. Designing Polygons PROBLEM 3.4 Arc of Learning Synthesis Abstraction Analysis NOW WHAT DO YOU KNOW? What information about a triangle allows you to draw a triangle? Draw a unique triangle? Explain why this true. What information about a quadrilateral allows you to draw a quadrilateral? Explain why this true. Key Terms Materials For each pair of students • Learning Aid 3.4A: Message Cards • Learning Aid 3.4B: Possible Polygon Designs • tools to create shapes: polystrips, angle rulers, dot paper, geoboards, geometry technology, etc. Pacing 1 day Groups 2 students A 16–21 C 33–37 E 42–43 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Earlier in the unit, students found that the sum of any two side lengths of a triangle is greater than the third side. They also found that given three side lengths that make a triangle, there is exactly one triangle. They learned that the sum of the angles of a triangle is 180°. In this problem, they use this knowledge to look at the conditions that produce unique triangles and use their knowledge of angles to create other polygons. PRESENTING THE CHALLENGE Launch the problem by asking students, “Suppose you want to text a friend to give directions for drawing an exact copy of the figure. What is the shortest message to do the job?” Have students brainstorm and share their ideas in their groups. There are five message cards on Learning Aid 3.4A. These could be divided up in the classroom with each partner group doing two or three message cards. Choose the message Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. 192 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Facilitating Discourse Teacher Moves LAUNCH After some class discussion of approaches, you could segue to the Initial Challenge. Guide your students by suggesting that they look for some general guidelines they could use for writing their messages about triangles with minimum text and maximum information. cards for each group based on the formative assessment you’ve been gathering on each student to meet their current needs. (Agency, Identity, Ownership) EXPLORE PROVIDING FOR INDIVIDUAL NEEDS As students work on the message cards, they should justify their answers. Encourage students to share their triangles on poster paper. You can use the posters in a gallery walk during the Summarize. At that time, students can add new information to their notes from the posters. Students could use polystrips, angle rulers, geoboards, dot paper, or geometry technology to determine if the triangles are possible. As students are finishing the Initial Challenge, give students cards from Learning Aid 3.4B: Possible Polygon Designs. Make sure each partner group gets at least one that is not possible. PLANNING FOR THE SUMMARY As you are circulating, take note of how students are justifying when triangles/ quadrilaterals are and are not possible. Have students share these generalizations in the summary. Look for students using angles and/or side lengths as their justifications. You can have them add this work to their posters for the gallery walk in the Summarize. SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Start with a gallery walk of student-created posters, or have groups share their findings with the whole class. Ask the students where they notice differences. Students should be able to resolve these differences. MAKING THE MATHEMATICS EXPLICIT Have students share thoughts, questions, and conclusions about the five message cards. Repeat a similar discussion with the What If . . . ? 12 designs. Here is a sample conversation from one classroom: Shandra: We thought there were three triangles for Message 1. (Shares the three they found, and they are the same triangle but oriented differently.) 46° 3.6 cm 4 cm 3 cm 60° 74° B C A 3 cm 3.6 cm 4 cm 60° 3.6 cm 4 cm 3 cm 60° Lewis: That is the same triangle but just turned on different sides. (Goes up to a poster and shows rotating the triangles to all be the same.) Shandra: Oh. Now I see. Arvin: We thought Message 4 was fun. Gallery Walk Claim, Support, Question 3.4 Problem 3.4 Designing Polygons 193 SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) Teacher: What was fun in Message 4? Arvin: Since we had all the angles for the triangle, we could create a lot of different triangles by just lengthening or shortening the sides once we had the angles set. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A Shandra: We found that same thing with Design 2 in What If . . . ? Situation A. We knew three angles and could find lots of triangles. Teacher: What does this tell us about triangles? Arvin: If we have three angle measures, we can make lots of triangles that have those angles. Shandra: I agree. But sometimes three angle measures do not form any triangles. We cannot just have any three angle measures. So if you have three angle measures that make a triangle, then there are many triangles possible. Teacher (writing on anchor chart): We have found that “Many triangles are possible if we have three angle measures that make a triangle.” As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Anchor Chart 3.4 194 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Problem Overview The objective of this problem is to develop student understanding of the conditions that determine unique shapes. This is the foundation for subsequent work on similarity and congruence later on. The storyline of the problem is a challenge to convey a lot of information with only a few words to produce a unique triangle. For example, if the three corresponding side lengths of two triangles are equal, the triangles are congruent (SSS Theorem). This property of congruence is what makes triangles rigid and useful in building. Similarly, if two angles and the included side are congruent to the corresponding angles and included side of another triangle, the triangles are congruent (ASA Theorem). Launch (Getting Started) Connecting to Prior Knowledge Earlier in the unit, students found that the sum of any two side lengths of a triangle is greater than the third side. They also found that given three side lengths that make a triangle, there is exactly one triangle. They learned that the sum of the angles of a triangle is 180°. In this problem, they use this knowledge to look at the conditions that produce unique triangles and use their knowledge of angles to create other polygons. Presenting the Challenge Launch the problem by asking students, “Suppose you want to text a friend to give directions for drawing an exact copy of the figure. What is the shortest message to do the job?” Have students brainstorm and share their ideas in their groups. After some class discussion of approaches, you could segue to the Initial Challenge. Guide your students by suggesting that they look for some general guidelines they could use for writing their messages about triangles with minimum text and maximum information. Implementation Note: There are five message cards on Learning Aid 3.4A: Message Cards. These could be divided up in the classroom with each partner group doing two or three message cards. Choose the message cards for each group based on the formative assessment you’ve been gathering on each student to meet their current needs. (Agency, Identity, Ownership) EXTENDED LAUNCH—EXPLORE—SUMMARIZE Extended Lanuch—Explore—Summarize 195 SAMPLE
• Message 1 gives a unique triangle. • Message 2 does not give a unique triangle. • Message 3 gives a unique triangle. • Message 4 does not give a unique triangle. • Message 5 gives a unique triangle. Distribute Learning Aid 3.4A: Message Cards or individual message cards. Explore (Digging In) Providing for Individual Needs As students work on the Initial Challenge, monitor their answers to see that they have reasons supporting their answers. As students work on the message cards, they should justify their answers. Encourage students to share their triangles on poster paper. You can use the posters in a gallery walk during the Summarize. At that time, students can add new information to their notes from the posters. (Portrayal) Implementation Note: As students are finishing the Initial Challenge, give them cards from Learning Aid 3.4B: Possible Polygon Designs to do What If . . . ? Situation A. Make sure each partner group gets at least one that is not possible. (Designs 3, 6, 8, and 11 are not possible.) You can have them add this work to their posters for the gallery walk in the summary. Students could use polystrips, angle rulers, geoboards, dot paper, or geometry technology to determine if the triangles are possible. Below is an example of using angle rulers to test Design 2 in Situation A. LES Design 2 Triangle with ∠A = 50º, ∠B = 100º, and ∠C = 30º Students set three angle rulers to the measures of 50º, 100º, and 30º. They put the angle rulers together to see if a triangle can be formed. Students notice that you can slide the ruler and form another triangle with those same three angle measures. 196 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
This shows that it is possible to make a triangle with the angle measures of 50º, 100º, and 30º. And the triangle created with those measure is not unique. Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? What information about a triangle allows you to draw a triangle? Draw a unique triangle? Explain why this true. What information about a quadrilateral allows you to draw a quadrilateral? Explain why this true. (As you are circulating, take note of how students are justifying when triangles/quadrilaterals are and are not possible. Have students share these generalizations in the summary. Look for students using angles and/or side lengths as their justifications.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies A fundamental question to be answered in this problem is “What minimum information about a triangle allows you to draw exactly one triangle?” A complete answer to this question is a major topic of high school geometry. At this point, students should have a strong sense that the answer is “At least three sides and/or angle measurements, but not any three.” (Time and Language) LES 0 1 2 3 4 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 100 110 120 130 150 140 160 170 170 160 150 140 130 120 110 100 90 80 70 60 50 30 40 20 10 0 1 2 3 4 5 6 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 100 110 120 130 150 140 160 170 170 160 150 140 130 120 110 100 90 80 70 60 50 30 40 20 10 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 100 80 110 120 130 140 150 160 170 170 160 150 140 130 120 110 100 80 90 70 60 50 40 30 20 10 0 1 2 3 4 5 6 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 100 80 110 120 130 140 150 160 170 170 160 150 140 130 120 110 100 80 90 70 60 50 40 30 20 10 100° 100° 50° 30° 30° Extended Lanuch—Explore—Summarize 197 SAMPLE
Start the summary with a gallery walk of student-created posters, or have groups share their findings with the whole class. Ask the students where they notice differences. Students should be able to resolve these differences. They should know that given three side lengths that form a triangle, there is only one triangle (SSS condition). When students are ready, discuss other conditions that produce unique triangles: ASA and SAS. Making the Mathematics Explicit The following is a conversation from one classroom while discussing the Initial Challenge message cards: Shandra: We thought there were three triangles for Message 1. (Shares the three they found, which are the same triangle but oriented differently.) 46° 3.6 cm 4 cm 3 cm 60° 74° B C A 3 cm 3.6 cm 4 cm 60° 3.6 cm 4 cm 3 cm 60° Lewis: That is the same triangle but just turned on different sides. (Goes up to a poster and shows rotating the triangles to all be the same.) Shandra: Oh. Now I see. Arvin: We thought Message 4 was fun. Teacher: What was fun in Message 4? Arvin: Since we had all the angles for the triangle, we could create a lot of different triangles by just lengthening or shortening the sides once we had the angles set. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A Shandra: We found that same thing with Design 2 in What If . . . ? Situation A. We knew three angles and could find lots of triangles. LES 198 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Teacher: What does this tell us about triangles? Arvin: If we have three angle measures, we can make lots of triangles that have those angles. Shandra: I agree. But sometimes three angle measures do not form any triangles. We cannot just have any three angle measures. So if you have three angle measures that make a triangle, then there are many triangles possible. Teacher (writing on anchor chart): Many triangles are possible if we have three angle measures that make a triangle. Have students share their thoughts, questions, and conclusions about the five message cards. Then repeat a similar discussion looking at the 12 designs in the What If . . . ? Suggested Questions • Do you have any What If . . . ? questions? • How did you decide if your polygon design was possible? (Answers will vary. Push students to discuss interior angle sums, side lengths, and shapes to pull out the vocabulary and concepts from the unit.) Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES Extended Lanuch—Explore—Summarize 199 SAMPLE
INITIAL CHALLENGE The drawing shows a triangle with measures of its angles and sides. Suppose you want to text a friend to give directions for making a drawing that is an exact copy of the figure with the shortest message possible. 46° 3.6 cm 4 cm 3 cm 60° 74° B C A • Which of these messages give enough information to draw a triangle that has the same shape and size as triangle ABC? Designing Polygons PROBLEM 3.4 Message 1 BC¯= 4 cm ∠B = 60° AB¯ = 3 cm Message 2 AB¯ = 3 cm BC¯ = 4 cm ∠C = 46° Message 3 ∠B = 60° BC¯ = 4 cm ∠C = 46° Message 4 ∠B = 60° ∠A = 74° ∠C = 46° Message 5 ∠B = 60° ∠C = 46° AC¯= 3.6 cm Message 1: There is enough information. Only triangle ABC is possible. The angle sets up different possible lengths for side AC. 60° 4 cm A 3 cm 3.6 cm B C 74° 46° Knowing the length of the two sides sets the length of the third side. 46° 3.6 cm 4 cm 3 cm 60° 74° B C A Answers Embedded in Student Edition Problems Answers 200 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Message 2: Not enough information. This does not create a unique triangle. We could create the given triangle. 46° 3.6 cm 4 cm 3 cm 60° 74° B C A But we could also “swing” side AB to get a different triangle where ∠A is now much smaller. 46° 3.6 cm 4 cm 3 cm 60° 74° B C A Message 3: There is enough information. Only triangle ABC is possible. The angles set up possible triangles. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A Then the side length of BC establishes the given triangle. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A Message 4: Not enough information. We will have the correct angle sizes. Many triangles are possible. We will have the correct angle sizes. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A But the side lengths could be different. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A Message 5: There is enough information. Only triangle ABC is possible. We know the measure of two angles and that the sum of the angles must be 180°. So we would know the measure of the third angle. ∠A + 60° + 46° = 180° So this is like Message 3. The angles set up possible triangles. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A The side length of AC establishes the given triangle. 74° 60° 46° 3 cm 4 cm 3.6 cm B C A 3.4 Answers Problem 3.4 Designing Polygons 201 SAMPLE
WHAT IF . . . ? Situation A. Designing Polygons For each design, do the following: 1. Decide if the criteria can be used to design the polygon. 2. If it is not possible, explain why. 3. If so, make a sketch of the polygon. Label the angle and side measurements. Is the polygon unique? Explain. Design 1 Triangle ∠A = 40º AB¯ = 6 cm AC¯= 6 cm Design 2 Triangle ∠A = 50º ∠B = 100º ∠C = 30º Design 3 Triangle AB¯ = 5 cm BC¯= 2 cm AC¯= 10 cm Design 4 Triangle AB¯ = 12 cm ∠A = 60º ∠B = 40º There is one possible triangle with these measurements. 6 cm 6 cm 4 cm 40° A triangle with the given angle measures is not unique. Many triangles can be made with the angle measures. We can change all the side lengths in the same way to make another. B C 10 cm A 16 cm 20 cm 30° 50° 100° B C 5 cm A 8 cm 10 cm 30° 50° 100° No, we cannot make a triangle. 5 + 2 < 10 So the two smaller lengths will not touch the longer length. There is one possible triangle with these measurements. 60° 40° A C B 12 cm 11 cm Design 5 Right Triangle The angle measures are x, 2x, and 3x. Design 6 Right Triangle The angle measures are x, 3x, and 3x. Design 7 Right Triangle The angle measures are x, __1 2 x, and __1 2 x. Design 8 Quadrilateral ∠A = 90º ∠B = 60º ∠C = 60º ∠D = 100º 3.4 Answers 202 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
This will be a right triangle. x + 2x + 3x = 180 6x = 180 x = 30 2x = 60 3x = 90 A C B 3x 2x x There are many triangles possible. The angles would all be 30-60-90, but the side lengths could change. x A C B 3x 3x 2x 2x This will not be a right triangle. It will be isosceles. x + 3x + 3x = 180 25.7 + 77.1 + 77.1 ≈ 180 A triangle with these angle measures is possible. It is difficult to be that precise when sketching the triangle. It would be a tall isosceles. 3x 3x x We are only focused on angle measure, so this triangle is not unique. x x 3x 3x 3x This will be a right triangle. x + __1 2 x + __1 2 x = 180 90 + 45 + 45 = 180 With two angles the same measure and a right angle, this is an isosceles right triangle. x x 1 2 x 1 2 We are only focused on angle measure, so this triangle is not unique. x x x 1 2 x 1 2 x 1 2 Not possible. The angle measures of a quadrilateral add up to 360°. We are only given 310° in the given measures. Design 9 Hexagon ∠A = 120º ∠B = 120º ∠C = 120º ∠D = 120º ∠E = 120º ∠F = 120º Design10 Polygon A polygon that requires exactly two diagonals to make it rigid Design 11 Intersecting Lines Two intersecting lines with one pair of vertical angles with a measure of 60º and the other pair with a measure of 100º Design 12 Quadrilateral A quadrilateral with exactly one pair of parallel sides This would make a regular hexagon with all the angles and sides equal measures. More than one hexagon can be created. This would be a pentagon. Not possible. The sum of the vertical angles of intersecting lines must be 360° (just like the angles on shapes that tile). 60 + 60 +100 + 100 ≠ 360 This describes a trapezoid. 3.4 Answers Problem 3.4 Designing Polygons 203 SAMPLE
NOW WHAT DO YOU KNOW? What information about a triangle allows you to draw a triangle? Draw a unique triangle? Explain why this true. What information about a quadrilateral allows you to draw a quadrilateral? Explain why this true. Unique Triangle Many Triangles No Triangles When we are given: • all three side lengths; • angle measures and one side length (because we can find the third angle measure); or • two side lengths with the measure of the angle that joins the two sides or an angle measure and the lengths of the sides that form the rays of the angle. When we are given all three angle measures. This also works when given just two angles’ measures. We can find the third angle measure because we know that the angle sum of a triangle is 180°. When the sum of the two shorter sides is less than the longest side. Short side + Short side < Long side For quadrilaterals, we know that the • angle measures must add up to 360°; and • the longest side cannot be larger than the sum of the three other sides. 120° 120° 120° 120° 120° 120° 5 cm 5 cm 5 cm 5 cm 5 cm 5 cm 120° 120° 120° 120° 120° 120° 2 cm 2 cm 2 cm 2 cm 2 cm 2 cm Or adjacent angles must be 180° if they are formed from two lines intersecting. 80° 100° Parallelograms (including squares and rectangles) have two pairs of parallel sides. 3.4 Answers 204 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Problem 3.4 Designing Polygons 205 Name Date Class Message 1 BC¯= 4 cm ∠B = 60° AB¯ = 3 cm Message 2 AB¯ = 3 cm BC¯ = 4 cm ∠C = 46° Message 3 ∠B = 60° BC¯= 4 cm ∠C = 46° Message 4 ∠B = 60° ∠A = 74° ∠C = 46° Message 5 ∠B = 60° ∠C = 46° AC¯= 3.6 cm Message Cards 3.4A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
206 Investigation 3 Designing Polygons: The Angle Connection Name Date Class Situation A Design 1 Triangle ∠A = 40º AB¯ = 6 cm AC¯= 6 cm Design 2 Triangle ∠A = 50º ∠B = 100º ∠C = 30º Design 3 Triangle AB¯ = 5 cm BC¯= 2 cm AC¯= 10 cm Design 4 Triangle AB¯ = 12 cm ∠A = 60º ∠B = 40º Design 5 Right Triangle The angle measures are x, 2x, and 3x. Design 6 Right Triangle The angle measures are x, 3x, and 3x. Design 7 Right Triangle The angle measures are x, __1 2 x, and __1 2 x. Design 8 Quadrilateral ∠A = 90º ∠B = 60º ∠C = 60º ∠D = 100º Design 9 Hexagon ∠A = 120º ∠B = 120º ∠C = 120º ∠D = 120º ∠E = 120º Design10 Polygon A polygon that requires exactly two diagonals to make it rigid Design 11 Intersecting Lines Two intersecting lines with one pair of vertical angles with a measure of 60º and the other pair with a measure of 100º Design 12 Quadrilateral A quadrilateral with exactly one pair of parallel sides Possible Polygon Designs LEARNING AID 3.4B © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
At a Glance What do you know about geometric shapes? The Mathematical Reflection provides an opportunity to discuss the goals of the Investigation. Students can pull together their reasoning from the Now What Do You Know questions to summarize their learning over the time. Students can record their responses to the Mathematical Reflection to create a record of their current understandings of the big ideas of the Unit. The Mathematical Reflection can provide a self-assessment for students. Each student can have checkpoints of their understanding of the mathematics after each Investigation. Students can use the Mathematical Reflection to consolidate their mathematical thinking, take notes, and provide evidence of what they know and can do. A teacher can gain an understanding of student thinking during a discussion of the reflection question. Then one can assess individual understanding based on each student’s written work. For more on the Teacher Moves listed here, refer to the General Pedagogical Strategies section in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves EXPLORE Having students refer to their notes from the Now What Do You Know? in each problem in the Investigation can help them to synthesize all their ideas around geometric shapes. As a class, discuss the Mathematical Reflection. Use an idea like those on the next page to have students synthesize and record their thinking. Suggested Questions • After this investigation, what do we know about geometric shapes? • What did we learn in each problem of this investigation? • How might we describe the “big mathematical idea(s)” of the investigation? Time Anchor Chart Portrayal Language Mathematical Reflection MR Arc of Learning Analysis Synthesis Analysis Pacing __1 2 day Mathematical Reflection 207 SAMPLE
Student Responses At the beginning of the year, students will need more collaboration to outline and summarize the important ideas. They may need examples of writing, diagrams, and/or justifications from other students to help build their vision of what is expected when answering a Mathematical Reflection. Early in the year, you may want to start writing Mathematical Reflections as a whole group. Then as the year progresses, move to small groups, pairs, and finally individuals. Each investigation contributes to students’ conceptual understandings of the ideas in the unit. Students’ explanations at the beginning of a unit might be just forming. As you progress through the unit, students can use the contexts, representation, and connections to express a more solid understanding. By the end of the unit, students can create a complete picture of understanding. Example Strategies for Student Participation Here are a few creative strategies teachers use to encourage students’ ownership of their learning. Anchor Charts • After a discussion, chart the emerging understanding, and post it in the classroom. This can be done on poster paper or electronically. • Work with students throughout the unit to reference, add to, or refine their understandings. Note: For teachers who move classrooms or have multiple classes of the same grade level, create the chart in all classes, but keep just one to represent all of your classes. Post this one in the room, or bring it out when needed. Note Organization • Some teachers use the Mathematical Reflections as an organizer for note-taking during the investigation. • As part of the Summarize section of the problems, students record key ideas to the Now What Do You Know? reflection questions on a separate paper. • At the close of the investigation, students synthesize their notes into responses that summarize their emerging understandings of the ideas in the unit. Word Bank • As a class, create a word bank of terms from the Investigation. • Have groups of students write three or four statements using the words from the bank. • After formatively assessing their statements, you may choose to have a class discussion to refine the statements. Chalk Talk With a chalk talk, your writing does “the talking” instead of talking aloud. • Students post the question(s) on sheets of chart paper or on sections of your board. • Small groups record responses while collaborating in “chalk talk” format. • Students move to others’ work and add their thinking in the form of new ideas and connections. Final Reflection Presentation Teachers sometimes use the Mathematical Reflection after the last investigation as a summary of students’ learning. • Students consolidate their learning from the unit. • Teachers choose from various ways to present their ideas. Presentation choices might include creating a poster, written paper, presentation, or song/rap. Partner Write • Students create a written response to the reflection question with a partner. • Students discuss the reflection question with a partner. • Students create and write a response with a partner. 208 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
In this unit, we are investigating some general properties of geometric shapes (figures), including angles and polygons, and using this information to design shapes and solve problems. At the end of this investigation, ask yourself: What do you know about geometric shapes? Geometric shapes have special relationships between side lengths and angle measures. Some angle relationships include: Supplementary angles are two angles whose sum is 180°. Complementary angles are two angles whose sum is 90°. Vertical angles have the samesize measures. Knowing a few benchmark angles can help to estimate angle size or check accuracy when using a measuring tool. The sum of the interior angles of polygons changes in a predictable pattern depending on the numbers of sides. The relationship can be represented as S = 180(n − 2) or S = 180n − 360. The sum of the exterior angles of any polygon is always 360°. To tile without gaps or overlaps, the angles of the shapes must meet to form 360°. So the shapes can cover the area around a point. Some side relationships include: To form a polygon, each side must be less than the sum of the others. Only one triangle can be formed when the side lengths meet to form a triangle. (To form the triangle, the two smaller sides must be greater than the third side.) Unlike triangles, different quadrilaterals can be constructed with four given sides of the quadrilateral. For example, if you have two pairs of equivalent side lengths, you can form a rectangle by putting the equivalent lengths opposite each other. Or you can form a kite by putting the equivalent lengths adjacent to each other. Triangles are rigid. This is the reason triangles are used in buildings and other structures. Triangles can be formed inside other shapes to add support and increase the rigidity of a structure. MR Answers Embedded in Student Edition Problems Mathematical Reflection Mathematical Reflection 209 SAMPLE
Some angle and side relationships in triangles include being able to form one, more than one, or no triangles when given certain conditions. Unique Triangle Many Triangles No Triangles When we are given: • all three side lengths; • angle measures and one side length (because we can find the third angle measure); or • two side lengths with the measure of the angle that joins the two sides or an angle measure and the lengths of the sides that form the rays of the angle. When we are given all three angle measures. This also works when given just two angles’ measures. We can find the third angle measure because we know that the angle sum of a triangle is 180°. When the sum of the two shorter sides is less than the longest side. Short side + Short side < Long side 210 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
INVESTIGATION 3 APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) APPLICATIONS 1. Multiple Choice Which of the following combinations will not tile a flat surface? A. regular heptagons B. equilateral triangles C. regular hexagons D. squares Answer: A 2. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. regular octagons C. regular pentagons and regular hexagons D. regular hexagons and equilateral triangles Answer: D 3. Multiple Choice Which of the following combinations will tile a flat surface? A. regular heptagons and equilateral triangles B. squares and regular octagons C. regular pentagons and regular hexagons D. regular hexagons and squares Answer: B 4. A right triangle has one right angle and two acute angles. Without measuring the angles, what is the sum of the measures of the two acute angles? Explain your reasoning. The sum of the acute angles is 90°. The triangle angle sum (180°) minus a right angle (90°) is 90°. Answers Embedded in 5. Without measuring, find the measure of the angle labeled x in each regular polygon. a. x b. x a. regular hexagon: 6x = 720, x = 120° b. regular dodecagon: 12x = 1,800, x = 150° Applications—Connections—Extensions (ACE) 211 SAMPLE
ACE a. 90° 30° x c. 93° 135° 70° x b. 45° 45° x d. 60° 60° x x e. 37° 120° x g. This figure is a parallelogram. f. This figure is a regular hexagon. h. This figure is a trapezoid. a. 30 + 90 + x = 180. x = 60° b. 45 + 45 + x = 180. x = 90° c. 135 + 70 + 93 + x = 360. x = 62° d. 60 + x + 60 + x = 360. x = 120° e. x + 37 + 120 = 180. x = 23° f. 720 ÷ 6 = 120. x = 120° g. x = 70° vertical angles h. 67 + x + 90 + 90 = 360. x = 113° 120° x 67° x 70° x 6. Write an equation, and find the measure of each angle labeled x. 212 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
7. The following figure is a regular dodecagon. It has 12 sides. a. What is the sum of the measures of the angles of this polygon? 1,800° b. What is the measure of each angle? 150° c. Can copies of this polygon be used to tile a flat surface? Explain. This figure will not tile a flat surface because no integer multiple of 150 equals 360, the angle sum at each vertex of a tiling pattern. 8. Kele claims that the angle sum of a polygon he has drawn is 1660°. Can he be correct? Explain. The number 1,660 is not an integer multiple of 180, so it cannot be correct for the angle sum of any polygon. 9. Find the measure of each internal angle and the sum of the internal angles for the following concave polygons. a. b. c. a. 35° 35° 75° 75° 250° 250° 2 • 75 + 2 • 250 + 2 • 75 = 720 b. 270° 270° 90° 90° 90° 90° 90° 90° 6 • 90 + 2 • 270 = 1,080 ACE Applications—Connections—Extensions (ACE) 213 SAMPLE
c. 110° 110° 240° 145° 145° 75° 75° 2 • 110 + 2 • 145 + 2 • 75 + 240 = 900 10. A figure is called a regular polygon if all sides are the same length and all angles are equal. List the members of the Shape Set that are regular polygons. Regular polygons include A, B, C, D, E, and F. 11. Suppose in-line skaters make one complete lap around a park shaped like the quadrilateral below. GPS Route Menu c b a d 40° 76° 140° 104° Hardware Coffee Bagels Bank What is the sum of the angles through which they turn? 360° 12. Suppose in-line skaters complete one lap around a park that has the shape of a regular pentagon. a. What is the sum of the angles through which they turn? b. How many degrees will the skaters turn if they go once around a regular hexagon? A regular octagon? A regular polygon with n sides? Explain. a. 360° b. The answer to all questions is 360° because they will make exactly one complete turn as they skate around any polygon ACE 214 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
13. A class was asked what convinced them that the sum of exterior angles in any polygon is 360°. Here are three different points of view. "We were convinced when we drew a bunch of different figures and used my angle ruler to measure the exterior angles. They all came out close to 360°." "We were convinced when we thought about walking around the figure and realized that we made one complete turn or 360°." "We used the results about sums of interior angles and the fact that the measure of each interior angle plus its adjacent exterior angle is 180° to deduce the formula using algebra." What are the pros and cons of each argument? The aim of this task is to get students to reflect on mathematical practices that provide justification for discoveries. There is something to be said for the strategies proposed as convincing by each group—empirical evidence is very often the first step toward a generalization, and for most people, it provides practical intuitive confidence in a result. The limitation is, of course, that drawing and measurement are inexact and only give inductive evidence of what might be true. The thinking that infers an angle sum of 360° from imagining the turns that one makes in a complete circuit of a polygon is both intuitively appealing and logically convincing. The argument from a prior result about interior angles and deduction of connected exterior angles is perhaps more mathematically logical (assuming that the interior angle result is well established), but the path from the starting point to the result about exterior angles requires some fairly abstract and formal reasoning (sum of interior and exterior angles is n(180); sum of interior angles is 180(n − 2); so sum of exterior angles is 2(180)). This last reasoning is, of course, only as valid as the prior result. 14. Find the measures of the angles represented by a, b, c, and d. Explain your reasoning. 90° 60° 75° a b c d Students may reason in different orders. One solution might be: ∠a is vertical to the 90° angle. So ∠a = 90°. ∠b and the 75° angle are supplementary ∠b + 75° = 180° That makes ∠b = 105°. We know the measures of angles a and b. We know ∠a+ ∠b + ∠c + 60° = 360°. So 90° + 105° + ∠c + 60° = 360°. ∠c = 360° − 90° − 105° − 60° ∠c = 105° ACE Applications—Connections—Extensions (ACE) 215 SAMPLE
15. Are any of the labeled angles vertical? Supplementary? Complementary? Explain why. 90° 60° 75° a b c d Vertical: angles 90° and a, c and d. Each of these angles are “opposite” each other where two lines cross. Note that there are 16 possible pairs of vertical angles, but only 2 are labeled on the drawing. Supplementary: b and 75°. These angles add to 180°. Note that there are many possible pairs of supplementary angles, but only 1 is labeled on the drawing. Complementary: None of the given (labeled) angles are complementary (add to 90°). 16. If possible, draw the triangle described. Explain if the triangle is unique or if many triangles can fit the given information. a. A triangle with side = 2 in., side = 1 in., and ∠A = 75°. One triangle is possible. A triangle with side = 2 in., side = 1 in., and ∠BAC = 75° will look like this: C A B b. A triangle with ∠A = 75° and ∠C = 75°. There are many triangles that have ∠BAC = 75° and ∠ACB = 75°. All are similar to this with sides of different length. C A B c. A triangle with angles 45° and 60° and one side of length 2 in. There are many possible triangles with angles 45° and 60° and one side of length 2 in. depending on where you locate the length of 2. One possible answer is: C A B ACE 216 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
d. A triangle with side KL = 1 in., side LM = 1.5 in., ∠L = 135°, and side KM = 1 in. This cannot be constructed because once sides KL and LM are drawn with the given conditions, the distance from K to M is greater than 1 in. The given measurements do not fit the triangle inequality (1 + 1 > 1.5). 1 1.5 L M K 1 135° e. Triangle with all sides of length 1.5 in and all angles of 60°. One triangle is possible with all sides of length 1.5 in. and all angles of 60°. C A B 17. Draw the polygons described. a. A trapezoid PQRS. ∠P = 45°. ∠Q = 45°. Side = 1 in. Side = 2 in. A trapezoid PQRS that has ∠QPS = 45°, ∠RQP = 45°, side = 1 in., and side = 2 in. will look like this: P S R Q b. A parallelogram ABCD with two sides of length 2 in., two sides of length 1 in., and angles of 60° and 120° A parallelogram with two sides of length 2 in., two sides of length 1 in., and angles of 60° and 120° will look like this: A D C B c. A quadrilateral EFGH. Side = 1 in. Side = 2 in. Side = 3 in. Side = 6 in. Not possible. The sum of the three shortest side lengths must be longer than the longest side length. ACE Applications—Connections—Extensions (ACE) 217 SAMPLE
18. Which of these descriptions of a triangle ABC are directions that can be followed to draw exactly one shape? a. AB¯ = 2.5 in., AC¯ = 2 in., ∠B = 40° b. AB¯ = 2.5 in., AC¯ = 1 in., ∠A = 40° c. AB¯ = 2.5 in., ∠B = 60°, ∠A = 40° d. AB¯ = 2.5 in., ∠B = 60°, ∠A = 130° a. Two different triangles are possible. b. The triangle is unique by SAS. c. The triangle is unique by ASA. d. No triangle is possible because the sum of the measures of the angles exceeds 180°. 19. In the parallelogram, find the measure of each numbered angle. 4 1 117° 3 2 5 Angles 1, 3, and 5 are all 63°; angles 2 and 4 are both 117°. 20. Which of the following statements are true? Be able to justify your answers. a. All squares are rectangles. True b. No squares are rhombuses. False c. All rectangles are parallelograms. True d. Some rectangles are squares. True e. Some rectangles are trapezoids. False f. No trapezoids are parallelograms. True. Note: By our chosen definition, a trapezoid is a quadrilateral with one and only one pair of parallel sides. g. Every quadrilateral is a parallelogram, a trapezoid, a rectangle, a rhombus, or a square. False 21. Suppose you want to build a triangle with three angles measuring 60°. a. What do you think must be true of the side lengths? All the side lengths would be equal. This would be a regular triangle. b. What kind of triangle is this? equilateral c. Would this be a unique triangle? No, there are many possible triangles with all angles measuring 60˚. ACE 218 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
CONNECTIONS 22. Arun writes the equation d = 6t to represent the distance in miles, d, that riders could travel in t hours at a speed of 6 miles per hour. Make a table that shows the distance traveled every hour, up to 5 hours, if riders travel at this constant speed. Time (h) Distance (mi) 0 0 1.0 6 2.0 12 3.0 18 4.0 25 5.0 30 23. Use the equations to fill in the quantities in a copy of the table. a. m = 100 − k k 1 2 5 10 20 50 m 99 98 95 90 80 50 b. d = 3.5t t 1 2 5 10 20 40 d 3.5 7 17.5 35 70 140 24. The product of two numbers is 20. Find the value of n. a. n • 2 __1 2 = 20 8 b. 1 __1 4 • n = 20 16 c. n • 3 __1 3 = 20 6 ACE Applications—Connections—Extensions (ACE) 219 SAMPLE
25. The coordinate grid below shows four polygons. 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 11 12 y x 4 3 2 1 a. Give the coordinates of all vertices of each polygon. 1: (1, 7), (5, 11), (5,7) 2: (7, 8), (8, 10), (10, 10), (11, 8) 3: (6, 1), (6, 4), (10, 0), (10, 3) 4: (1, 2), (1, 5), (3, 2), (3, 5) b. Use the coordinates to find the lengths of as many sides as you can. 1: horizontal = 4 units, vertical = 5 units 2: horizontal = 2 units, 4 units 3: vertical = 3 units, 3 units 4: horizontal = 2 units, 2 units, vertical = 3 units, 3 units c. Describe as precisely as possible each type of triangle or quadrilateral shown. 1: right triangle 2: trapezoid 3: parallelogram 4: rectangle 26. Find the area of each of the following figures. ACE a. (4 × 5) ÷ 2 = 10 square units b. (6 × 8) ÷ 2 = 24 square units c. 6 × 4 = 24 square units a. b. 8 cm 6 cm c. 220 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
27. Find the area and perimeter of each figure. a. 5 cm 5 cm 9 cm 10 cm A = 65 cm2 (5 × 9 + 5 × 4) P = 38 cm (9 + 5 + 5 + 5 + 4 + 10) b. 10 cm 4 cm 2 cm 4 cm 3 cm 6 cm 5 cm A = 36 cm2 (4 × 3) + (10 × 2) + (2 × 2) P = 36 cm (4 + 2 + 2 + 5 + 4 + 3 + 6 + 10) c. 9 in. 1 2 9 in. 1 2 8 in. 8 in. 9 in. P = (9 __1 2+ 8) × 2 = 35 in. A = 9 × 8 = 72 sq. in. d. 10 cm 7 cm 81 cm 2 8 cm 7 10 P = 8 __7 10+ 10 + 7 = 25 __7 10 cm A = (7 × 8 __1 2 ) ÷ 2 = 29 __3 4 sq. cm ACE Applications—Connections—Extensions (ACE) 221 SAMPLE
e. 13 cm 13 cm 12 cm 10 cm P = 10 + 13 + 13 = 36 cm A = (10 × 12) ÷ 2 = 60 sq. cm 28. Multiple Choice How many feet are in one yard? A. 12 feet B. 3 feet C. 9 feet D. 27 feet Answer: B Multiple Choice How many feet are in one inch? A. 12 feet B. 3 feet C. 9 feet D. __1 12 foot Answer D Multiple Choice How many feet are in one mile? A. 5,280 ft. B. 3 ft. C. 2,300 ft. D. 100 ft. Answer: A 29. Find the area of each polygon. a. = 1 square meter A = 48 m2 ACE 222 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
b. A = 8 m2 c. A = 8 m2 30. Multiple Choice Figure QSTV is a rectangle. The lengths QR and QV are equal. What is the measure of angle x? Q R S V T x A. 20° B. 45° C. 90° D. 120° Answer: B 31. Find the volume and surface area of each rectangular prism. a. 12 cm 12 cm 12 cm S.A. = 6(12 × 12) = 864 square centimeters V = 12 × 12 × 12 = 1,728 cubic centimeters b. 12 in. 3 in. 1 2 7 in. S.A. = 2(7 × 3 __1 2 ) + 2(12 × 3 __1 2 ) + 2(12 × 7) = 301 square inches V = 7 × 12 × 3 __1 2= 294 cubic inches ACE Applications—Connections—Extensions (ACE) 223 SAMPLE
32. Write an equation to represent each question. a. Zuri charges $12 per hour for babysitting in her neighborhood. What equation relates her pay for a job to the number of hours she works? p = 12h, where p is Zuri’s pay and h is the number of hours she works. b. A gasoline service station offers 20 cents off the regular price per gallon every Tuesday. What equation relates the discounted price to the regular price on that day? d = r − 0.20, where d is the discounted price per gallon and r is the regular price per gallon. c. What equation shows how the perimeter of a square is related to the length of a side of the square? p = 4s, where p is the perimeter and s is the side length. d. A middle school wants to have its students see a movie at a local theater. The total cost of the theater and movie rental is $1,500. What equation shows how the cost per student depends on the number of students who attend? C = _____ 1,500 n , where c is the cost in dollars per student and n is the number of students. 33. Draw a polygon with the given properties (if possible). Decide if the polygon is unique. If not, design a different second polygon with the same properties. a. a triangle with a height of 5 cm and a base of 10 cm Multiple triangles are possible. b. a triangle with a base of 6 cm and an area of 48 cm Multiple triangles are possible. c. a triangle with an area of 12 square centimeters Multiple triangles are possible. d. a parallelogram with an area of 24 square centimeters Multiple parallelograms are possible. e. a parallelogram with a height of 4 cm and a base of 8 cm Multiple parallelograms are possible. 34. Draw the rectangle described. If there is more than one (or no) shape that you can draw, explain how you know that. perimeter = 24 cm and side of 8 cm A rectangle that has perimeter 24 cm and one side 8 cm will look like this: 8 8 4 4 ACE 224 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
35. Multiple Choice A triangle has a base of 4 units and an area of 72 square units. Which of the following is true? A. These properties do not make a triangle. B. These properties make a unique triangle. C. There are at least two different triangles with these properties. D. The height of the triangle is 18 units. Answer: C 36. Multiple Choice Which of the following could not be the dimensions of a parallelogram with an area of 18 square units? A. base = 18 units, height = 1 unit B. base = 9 units, height = 3 units C. base = 6 units, height = 3 units D. base = 2 units, height = 9 units Answer: B 37. Determine whether the measures in each pair are the same. If not, tell which measure is greater. Explain your reasoning. a. 1 square yard or 1 square foot 1 yd2 is greater. It is 9 ft2. b. 5 feet or 60 inches They are the same length. 5 × 12 in. = 60 in. c. 12 meters or 120 centimeters 12 m is greater because 120 cm is 1.2 m or 12 m is 1,200 cm. d. 12 yards or 120 feet 120 ft is greater because 12 yd is 36 ft. e. 50 centimeters of 500 millimeters They are the same length. 50 cm = 500 mm f. 1 square meter or 1 square yard Possible answer: One square meter is greater because a meter is greater than a yard. ACE Applications—Connections—Extensions (ACE) 225 SAMPLE
EXTENSIONS 38. Copy and complete the table. Sort the quadrilaterals from the Shape Set into groups by name and description. Sides and Angles of Quadrilaterals Name Examples in the Shape Set All sides are the same length. rhombus B, K, V All sides are the same length, and all angles are right angles. square B All angles are right angles. rectangle B, G, H, J Opposite sides are parallel. parallelogram B, G, H, J, K, L, M, N, V Only one pair of opposite sides are parallel. trapezoid O, R, S, U 39. Multiple Choice Which equation describes the relationship in the table? n 0 1 2 3 4 5 6 c 10 20 30 40 50 60 70 A. C = 10n B. C = 10 + n C. C = 10 D. C = 10 + 10n Answer: D 40. The table begun here shows a pattern for calculating the measures of interior angles in regular polygons with even numbers of sides. Regular Polygons Number of Sides Measure of Interior Angle 4 __1 2 of 180° 6 __2 3 of 180° 8 __3 4 of 180° 10 12 a. What entry would give the angle measures for decagons and dodecagons? Are those entries correct? How do you know? b. Is there a similar pattern for regular polygons with odd numbers of sides? If so, what is the pattern? Answer: even-sided polygons ACE 226 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
a. Following the pattern, the measure of an interior angle of a regular decagon would be __ 4 5 of 180°, or 144°. For dodecagons (12 sides), the pattern predicts __5 6of 180°, or 150°. These entries are correct according to results from Problems 2.1 and 2.2. Also, one could do an algebraic derivation from the formula __________ (n − 2) • 180 n , but this is beyond what is reasonable to expect for even very capable CMP students at the start of Grade 7. Realizing that this pattern is observed for the even numbers n = 2k, we get: __________ (n − 2) • 180 2k = (2k − 2) • 180 ___________ 2k = 2(k − 1) • 180 ___________ 2k = __________ (k − 1) • 180 k b. In the case of odd numbers n = 2k + 1, we get: __________ (n − 2) • 180 n − (2k + 1 − 2) • 180 ______________ 2k + 1 = 2(k − 1) • 180 ___________ 2k + 1 This leads to the pattern in the following table: Regular Polygons Number of Sides Measure of Interior Angle 3 __1 3of 180° = 60° 5 __ 3 5of 180° = 108° 7 __ 5 7of 180° ≈ 128.57° 9 __7 9of 180° = 140° 41. Below are a quadrilateral and a pentagon with the diagonals drawn from all of the vertices. a. How many diagonals does the quadrilateral have? How many diagonals does the pentagon have? Any quadrilateral has 2 diagonals; any pentagon has 5 diagonals. b. Find the total number of diagonals for a hexagon and for a heptagon. Any hexagon has 9 diagonals; any heptagon has 14 diagonals. ACE Applications—Connections—Extensions (ACE) 227 SAMPLE
c. Copy the table below, and record your results from parts (a) and (b). Number of Sides 4 5 6 7 8 9 10 11 12 Number of Diagonals 2 5 9 14 20 27 35 44 54 Look for a pattern relating the number of sides and the number of diagonals. Complete the table. d. Write a rule for finding the number of diagonals for a polygon with n sides. Answer: D = n2 ______ − 3n 2 42. The following drawing shows a quadrilateral with measures of all angles and sides. 2.3 cm 3 cm 3.7 cm 4 cm A D B C 110° 75° 60° 115° Suppose you wanted to text a friend giving directions for drawing an exact copy of it. For each of the following short messages, tell whether it gives enough information to draw a quadrilateral that has the same size and shape as ABCD. a. AB¯ = 3 cm, BC¯ = 4 cm, CD¯ = 2.3 cm b. AB¯ = 3 cm, ∠B = 60°, BC¯ = 4 cm, ∠C =115°, ∠A = 110° c. AB¯ = 3 cm, ∠B = 60°, BC¯ = 4 cm, ∠C =115°, CD¯ = 2.3 cm Answers: a. The directions are inadequate. b. The directions will define exactly the shape that is expected. c. The directions will define exactly the shape that is expected. 43. In parts (a)–(d), write the shortest possible message that tells how to draw each quadrilateral so that it will have the same size and shape as those below. ACE a. c. b. d. 228 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Answers: a. Draw a square with sides 1 inch. b. Draw a rectangle with sides 1.25 inches and 0.5 inches. c. Draw a rhombus with sides 1 inch and one angle of 60°. d. Draw a quadrilateral with side AB = __3 4 in. ∠B = 135°. side BC = 0.5 in. ∠C = 115°. side CD = 1 __1 4 in. Note: Other correct directions are possible. e. What is the minimum information about a quadrilateral that will allow you to draw an exact copy? The quadrilateral has 4 sides, 4 angles, and 2 diagonals. A unique quadrilateral cannot be defined by 4 of the 10 elements. At least 5 elements are needed to draw a unique quadrilateral. Note: It is not possible to construct a unique quadrilateral with 1 side and 4 angles. ACE Applications—Connections—Extensions (ACE) 229 SAMPLE
Unit Test 1. A protractor is shown. a. What is the measure of each angle? ∠1 ∠2 ∠3 ∠4 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 4 0 3 2 1 b. Name two angles that are complementary. 2. A triangle has sides of 4 and 6 units. The measurement of the longest side is missing. Jade says that one possibility for the unknown side length is 11. Do you agree with Jade? Why or why not? Name Date Class 3. In the figure below the measure of ∠ 2 is 45°. a. Find the measure of ∠1 = _________. Explain. b. Find the measure of ∠3 = _________. Explain. c. Name a pair of supplementary angles. 5 8 6 3 4 1 2 7 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 230 Shapes and Designs SAMPLE
Name Date Class 4. Rectangle ABCD has a diagonal DB. A D x w B C a. Find the measure of the angle marked w. 56° Show your work. b. Hank writes an equation to show how to find the measure of angle w. Hank’s Equation: w + 56 = 180 Do you agree with his equation? Explain. c. Find the measure of the angle marked x. Show your work. d. Write an equation that shows someone else how you can find the measure of ∠x. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Unit Test 231 SAMPLE
5. Use the quadrilateral below. Find the measures of the angles. 4 3 1 2 105° 75° 40° 140° Name Date Class a. Measure of ∠1 is _________. b. Measure of ∠2 is _________. c. Measure of ∠3 is _________. d. Measure of ∠4 is _________. e. Explain how you know that the measures are correct without using an angle rule or protractor. 6. Consider each set of side and angle criteria. Circle the correct response to show that the criteria can create only one triangle, more than one triangle, or no triangles. Explain each of your answers. a. Triangle ABC with AB¯ = 3 cm BC¯ = 6 cm AC¯ = 8 cm Only one triangle More than one triangle None Explain: b. Right Triangle ABC with ∠A = 30° ∠B = 60° Only one triangle More than one triangle None Explain: © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 232 Shapes and Designs SAMPLE
c. Triangle ABC with ∠C = 45° AB¯ = 10 cm AC¯ = 15 cm Only one triangle More than one triangle None Explain: d. Triangle ABC with AB¯ = 3 cm BC¯= 6 cm AC¯= 10 cm Only one triangle More than one triangle None Explain: Name Date Class © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Unit Test 233 SAMPLE
Unit Test: Answers 1. A protractor is shown. a. What is the measure of each angle? ∠1 = 30° ∠2 = 60° ∠3 = 60° ∠4 = 30° 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 4 0 3 2 1 b. Name two angles that are complementary. Possible Answers ∠1 and ∠2 ∠4 and ∠3 ∠1 and ∠3 ∠4 and ∠2 2. A triangle has sides of 4 and 6 units. The measurement of the longest side is missing. Jade says that one possibility for the unknown side length is 11. Do you agree with Jade? Why or why not? Jade’s suggested length of 11 units is too large. The third side must have a length less than the other two lengths: 4 + 6 = 10. 3. In the figure below the measure of ∠ 2 is 45°. a. Find the measure of ∠1 = _________ Possible explanation: ∠1 is supplementary to ∠2. b. Find the measure of ∠3 = _________ Possible explanation: ∠3 is supplementary to ∠1. ∠3 = ∠2 because they are vertical angles. c. Name a pair of supplementary angles. Many possible answers. ∠1 and ∠2 ∠1 and ∠3 ∠1 and ∠6 ∠1 and ∠7 ∠2 and ∠4 ∠2 and ∠8 ∠2 and ∠5 ∠3 and ∠4 ∠3 and ∠8 ∠3 and ∠5 ∠4 and ∠6 ∠4 and ∠7 ∠5 and ∠6 ∠5 and ∠8 ∠6 and ∠8 ∠7 and ∠8 135° 45° 5 8 6 3 4 1 2 7 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 234 Shapes and Designs SAMPLE
4. Rectangle ABCD has a diagonal DB. A D x w B C a. Find the measure of the angle marked w. 56° Show your work. 34° Possible work: 180 − 90 − 56 = w or 56 + 34 = 90 b. Hank writes an equation to show how to find the measure of angle w. Hank’s Equation: w + 56 = 180 Do you agree with his equation? Explain. No. Possible explanation: Hank needs to add all 3 angles together to get 180° in the triangle. He is missing the 90° angle. w + 56 + 90 = 180 or We know that a triangle has an angle sum of 180°. Since this is a rectangle, we know that ∠c is 90°. So w + 56 = 90. c. Find the measure of the angle marked x. Show your work. 34° Possible work: 90 − 56 = 34 d. Write an equation that shows someone else how you can find the measure of ∠x. Possible answers 90 − 56 = x 56 + x = 90 90 − x = 56 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Unit Test: Answers 235 SAMPLE
5. Use the quadrilateral below. Find the measures of the angles. 4 3 1 2 105° 75° 40° 140° a. Measure of ∠1 is _________. b. Measure of ∠2 is _________. c. Measure of ∠3 is _________. d. Measure of ∠4 is _________. e. Explain how you know that the measures are correct without using an angle rule or protractor. Answers will vary. Possible answer: In each case, the interior and exterior angles are supplementary. So the interior angle is equal to 180° − exterior angle. 40° 75° 105° 140° 6. Consider each set of side and angle criteria. Circle the correct response to show that the criteria can create only one triangle, more than one triangle, or no triangles. Explain each of your answers. a. Triangle ABC with AB¯ = 3 cm BC¯ = 6 cm AC¯ = 8 cm Only one triangle More than one triangle None Explain: The sum of any two sides is longer than the other side. There is only one scalene triangle that is possible. b. Right Triangle ABC with ∠A = 30° ∠B = 60° Only one triangle More than one triangle None Explain: The sum of the three angles is 180°, which would make a triangle; however, the side lengths could be many different sizes. Three examples include: 90° 30° 60° © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 236 Shapes and Designs SAMPLE
c. Triangle ABC with ∠C = 45° AB¯ = 10 cm AC¯ = 15 cm Only one triangle More than one triangle None Explain: Different angles at A could be formed joining sides AB and AC. Example: B C A 10 cm 15 cm 45° C B A 15 cm 10 cm 45° d. Triangle ABC with AB¯ = 3 cm BC¯= 6 cm AC¯= 10 cm Only one triangle More than one triangle None Explain: The sum of any two sides is not longer than the other sides: 3 + 6 < 10. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Unit Test: Answers 237 SAMPLE
PROBLEM CORRECTIONS WITH COMMON CORE STATE STANDARDS OF MATHEMATICS AND MATHEMATICAL PRACTICES Students develop an understanding of important mathematical ideas by solving problems and reflecting on the embedded mathematics. Each problem encourages students to use and connect their new understandings to previous understandings and the real world. All the problems in this unit address one or more of the Common Core State Standards of Mathematics (CCSSM) (NAGC & CCSSO 2010). Since the students choose their solution paths, it is not always possible to predict which practices they will use. As a contextual problem-based curriculum, students will always be using MP1. In addition, they will use other practices. This table lists examples of possible practices students may use to solve the problem. Problem CCSSM Mathematical Practices 1.1 7.G.A.2 MP1 MP3 1.2 7.G.A.2 MP1 MP7 1.3 7.G.A.2 MP1 MP2 Mathematical Reflection 7.G.A.2 MP1 MP3 2.1 7.EE.B.4, 7.EE.B.4.A, 7.G.A.2 MP1 MP8 2.2 7.RP.A.1, 7.RP.A.2.A, 7.RP.A.2.C, 7.EE.B.4, 7.EE.B.4.A, 7.G.A.1 MP1 MP5 2.3 7.EE.B.3, 7.EE.B.4, 7.EE.B.4.A, 7.G.B.5 MP1 MP6 Mathematical Reflection 7.RP.A.1, 7.RP.A.2.A, 7.RP.A.2.C, 7.EE.B.3, 7.EE.B.4, 7.EE.B.4.A, 7.G.A.1, 7.G.A.2 , 7.G.B.5 MP1 MP3 3.1 7.G.A.2, 7.G.B.5 MP1 MP8 3.2 7.EE.A.2, 7.EE.B.3, 7.EE.B.4, 7.EE.B.4.A, 7.G.B.5 MP1 MP4 3.3 7.EE.A.2, 7.EE.B.3, 7.EE.B.4, 7.G.B.5 MP1 MP7 CORRELATIONS 238 Shapes and Designs SAMPLE